Laura Walton
2
Abstract
Surveillance is the first line of defence against disease, whether
to monitor endemic cycles
or to detect emergent epidemics. Knowledge of disease in wildlife
is of considerable
importance for managing risks to humans, livestock and wildlife
species. Recent public
health concerns (e.g. Highly Pathogenic Avian Influenza, West Nile
Virus, Ebola) have
increased interest in wildlife disease surveillance. However,
current practice is based on
protocols developed for livestock systems that do not account for
the potentially large
fluctuations in host population density and disease prevalence seen
in wildlife.
A generic stochastic modelling framework was developed where
surveillance of wildlife
disease systems are characterised in terms of key demographic,
epidemiological and
surveillance parameters. Discrete and continuous statespace
representations respectively,
are simulated using the Gillespie algorithm and numerical solution
of stochastic differential
equations. Mathematical analysis and these simulation tools are
deployed to show that
demographic fluctuations and stochasticity in transmission dynamics
can reduce disease
detection probabilities and lead to bias and reduced precision in
the estimates of
prevalence obtained from wildlife disease surveillance. This
suggests that surveillance
designs based on current practice may lead to underpowered studies
and provide poor
characterisations of the risks posed by disease in wildlife
populations. By parameterising the
framework for specific wildlife host species these generic
conclusions are shown to be
relevant to disease systems of current interest.
The generic framework was extended to incorporate spatial
heterogeneity. The impact of
design on the ability of spatially distributed surveillance
networks to detect emergent
disease at a regional scale was then assessed. Results show dynamic
spatial reallocation of a
fixed level of surveillance effort led to more rapid detection of
disease than static designs.
This thesis has shown that spatiotemporal heterogeneities impact on
the efficacy of
surveillance and should therefore be considered when undertaking
surveillance of wildlife
disease systems.
1.2 Wildlife disease and
conservation.........................................................................
14
1.4 Wildlife Disease Surveillance
.................................................................................
17
1.4.1 The Probability of Detection
.......................................................................
20
1.4.2 Estimating the True Prevalence: Bias and Standard Deviation
.................. 22
1.4.3 Improving wildlife disease surveillance
...................................................... 24
1.5 Modelling Wildlife Populations, Disease and Surveillance
................................... 25
1.5.1 Temporal Modelling of Wildlife Populations with Disease
........................ 26
1.5.2 Spatial Temporal Modelling of Wildlife Populations, Disease
and Surveillance
..........................................................................................................................
35
2.1 Abstract
.................................................................................................................
39
2.2 Introduction
...........................................................................................................
39
2.3 Methods
................................................................................................................
41
2.3.2 Statistics generated from the model
.......................................................... 42
2.3.3 Model
Implementation...............................................................................
42
4
2.4.2.1 Effect of host demography and transmission dynamics
.................. 51
2.4.2.2 Limits to disease detection in wildlife disease systems
................... 52
2.5 Discussion
..............................................................................................................
54
3. An Approach to Assessing Wildlife Disease Surveillance in Real
Systems: tools and
applications
............................................................................................................
57
3.3.2 Model
Implementation...............................................................................
61
3.4 Results
...................................................................................................................
65
3.4.1 Estimating disease prevalence in badger and rabbit
populations ............. 66
3.4.1.1 Effect of disease system
..................................................................
66
3.4.2 Detecting
disease........................................................................................
74
3.4.4 Imperfect diagnostic tests
..........................................................................
78
3.5 Discussion
..............................................................................................................
82
surveillance: impacts of emergent disease.
..............................................................
86
4.1 Abstract
.................................................................................................................
87
4.2 Introduction
...........................................................................................................
87
4.3 Methods
................................................................................................................
89
4.3.2 Model
Implementation...............................................................................
91
4.4 Results 4.4.1 Static surveillance: spatial distribution of
effort............................. 94
4.4.2 Static surveillance: stratified designs
......................................................... 96
4.4.3 Dynamic designs
.......................................................................................
101
5
5. General Discussion: Towards a new approach to wildlife disease
surveillance ..... 109
5.1 Background
..........................................................................................................
109
5.2 Fluctuations undermine wildlife disease surveillance
........................................ 110
5.3 Tools for assessing wildlife disease surveillance in real
systems ........................ 112
5.4 Spatial heterogeneity and the design of wildlife disease
surveillance ............... 113
5.5 Future Work
........................................................................................................
114
1.1 Relationship between discrete and continuous (SDE) statespace
model
implementations
..................................................................................................
119
2.1.1.1 Discrete to continuous time
..........................................................
128
2.1.1.2 Translating from multi to singlestage models
............................. 129
2.1.2 Estimating Rabbit Parameters
..................................................................
133
Appendix 2.2
.........................................................................................................
135
List of Tables
1.1 SI model with rate and effects for each event type 29
1.2 Expectations and variancecovariance rates 33
1.3 Expectation and variancecovariance 34
2.1 Model structure 43
4.1 Event, Rate and Effect on the State Space of the model 92
7
List of Figures
1.1 Effect of disease prevalence on the sample size required to
detect disease 22
1.2 Using binomial theory to estimate the expected bias and std dev
in the
surveillance estimate of prevalence 24
1.3 An example of a logistic growth curve 28
1.4 Deterministic and Stochastic Time Trajectories 29
2.1 Effect of host demography and disease transmission 46
2.2 Effect of surveillance design 48
2.3 Effect of hostpathogen and surveillance dynamics on probability
of
detection 52
2.4 Fluctuations reduce power to detect disease 53
3.1 Trace plots from SDE and Gillespie implementation runs of the
model 64
3.2 Effect of infected population stability and disease
transmission on
surveillance in Badgers 68
3.3 Effect of infected population stability and disease
transmission on
surveillance in Rabbits 70
3.4 Effect of disease transmission and infected population death
rate on
surveillance 71
3.5 Effect of surveillance design 72
3.6 Effect of surveillance design 73
3.7 Effect of hostpathogen and surveillance dynamics on the
probability of
detection 75
3.8 Effect of trappability and surveillance dynamics in Badgers
77
3.9 Effect of test sensitivity and disease dynamics in Badgers
79
3.10 Effect of disease dynamics and test sensitivity in Badgers
81
4.1 Effect of the distribution of surveillance effort between
patches 95
4.2 The effect of habitat quality on stratified sampling schemes
97
4.3 The effect of the suitability of less favourable on
surveillance 99
8
4.4 The effect of surveillance effort and secondary transmission
100
4.5 Effect of switching 102
4.6 Switching and deployed effort 103
4.7 The effect of between patch transmission for different
switching levels 104
9
Acknowledgements
Firstly, I would like to thank my supervisor’s Glenn Marion and
Mike Hutchings for taking a
leap of faith on a 22 year old girl about to graduate with an
undergraduate degree in
statistics. It has been a privilege to work so closely with two
people of such knowledge and
expertise. Glenn’s relentless comments on my many redrafts have
been invaluable in this
process. He has been my main supervisory support, I’ve always been
able to go to him with
any problem, and have never felt that any question was too simple.
Mike’s knowledge of
ecology and his ability to relate my theoretical results to real
systems has been instrumental
in developing this research. His enthusiasm for this work has never
faltered, which has had
a positive influence on my own motivation and enthusiasm! This
whole process has been a
learning curve in more ways than one, and I couldn’t have done it
without the support of
both Glenn and Mike.
Secondly, I would like to thank Ross Davidson, for his time,
patience and expertise in C++.
His guidance and problem solving skills have been extremely helpful
and have aided my
programming development, which is a skill that I will carry
forward. I would also like to
thank my supervisor Piran White for always being happy to help,
answering questions and
providing feedback. His independent perspective has helped greatly
in formulating
arguments and developing key messages.
Thirdly I would like to thank some important people in my life. My
parents have been the
biggest rock, always believing in me, even at my lowest points.
They have listened to my
worries, my joy, put up with my bad moods and took an interest in
my theories and results,
all with love and support. They have helped me above and beyond and
I couldn’t ask for a
better family. I would also like to thank my friend Rachael for
having an ear to listen and a
shoulder to cry on! Rachael has celebrated the good times with me
and distracted me from
the bad, and has always found words of encouragement when I needed
them most.
I would also like to thank Kevin Kane for employing me, having an
income has come in very
useful in my fourth year! He has been very understanding and
flexible with my working
hours in these hectic last few months, and I am truly grateful.
Finally, thank you to Team
AML for their friendship and humour which have made my transition
to London so much
easier. They have told me time and again that I should hurry up and
finish my PhD so I can
join them in cocktails during happy hour. Now that time has come,
and I say cheers to that!
10
Author’s Declaration
I declare that this thesis is a presentation of original work and I
am the sole author. This
work has not previously been presented for an award at this, or any
other University. All
sources are acknowledged as references.
11
Introduction
Wildlife diseases have the potential, not only to impact greatly on
the populations of
wildlife species themselves, but also on human and livestock
populations. A demonstrative
example of how detrimental zoonoses can be to human health and the
economy is the
recent swine flu pandemic in 2009, which has origins in both
domestic pigs and wild boar
(Shoham 2011). The disease outbreak started in Mexico and the USA
and quickly spread
worldwide followed by vast media coverage. The official worldwide
death toll according to
the World Health Organisation (WHO) as of 28th March 2010 was
17,483 (Girard et al.
2010). As well as the impact on human health, the economic impact
of Mexico alone was
estimated as > $3.2 billion (Girard et al. 2010). Wildlife
diseases also have the opportunity
to affect biodiversity and conservation efforts (Daszak et al.
2000; Smith et al. 2009; Pagán
et al. 2012). Mathematical modelling is a tool which can be (and is
widely) used to simulate
disease systems (Renshaw 1991; Keeling & Rohani 2007) by
manipulating equations in order
to gain insights into the behaviour of the modelled system.
Mathematical modelling will be
used in this thesis in such a way to understand the effect that
host fluctuations, disease
dynamics and spatial heterogeneities have on the efficacy of
wildlife disease surveillance.
The literature surrounding wildlife disease is reviewed in the next
sections and the impact it
has on human health, wildlife health and conservation and the
economy is outlined to show
the importance of wildlife disease research.
1.1 Wildlife diseases and Humans
There is an increasing understanding in the literature that
wildlife diseases pose a threat to
human health (Bengis et al. 2004; Belant & Deese 2010; Kuiken
et al. 2011). The World
Organisation for Animal Health (OIE) stated in an editorial that
‘Surveillance of wildlife
diseases must be considered equally as important as surveillance
and control of diseases in
domestic animals’ (Vallat 2008). In a comprehensive literature
review carried out by Talyor
et al, it was reported that out of all the infectious organisms
known to be pathogenic to
12
humans, 61% were zoonotic. It was also found that out of the 175
diseases considered to be
“emerging” (Lederberg et al. 1992), 75% were zoonotic (Taylor et
al. 2001). This was higher
than expected by the authors and is indicative of the importance of
wildlife disease
research in terms of predicting and controlling emerging outbreaks,
and promoting human
health and safety. In comparison with human and livestock systems,
with wildlife disease
there are many added complications in terms of population
demography and habitat
location, and even though momentum is building behind wildlife
disease research it is still
the most poorly understood (Jones et al. 2008).
There are many examples of zoonotic diseases in humans for which
wildlife species act as
an intermediary for disease transmission. Nipah is an RNA virus
initially detected in pigs in
Malaysia in 1999 (when at the same time it appeared that pig
farmers were suffering from
an outbreak of viral encephalitis), it is closely related to the
Hendra Virus discovered in 1994
in Australia. Host infection by Nipah virus is associated with a
marked respiratory and
neurological syndrome which can be followed by the sudden death of
pigs. In the later
stages of the initial outbreak of this disease, Nipah was
characterised as causing a high
mortality rate in humans when it emerged that the same causative
agent was to blame for
both the pig and pig farm workers mysterious illnesses. The
outbreak had devastating
consequences for the Malaysian Peninsular’s pig farming industry
with an overall loss of
1.08 million pigs and a reduction in pig farms from 1885 to 829
(Nor et al. 2000). Although
the catalyst of the original outbreak of Nipah virus in pig farmers
was domestic pigs (Chua et
al. 1999), the natural host for the disease is fruit bats. The main
drivers associated with the
spread of this virus were identified as deforestation, drought and
the expansion of pig
farming in Malaysia. Fruit bats were forced to move out of their
natural habitat as a result
of depleting resources available to them, and nearby agricultural
areas with productive fruit
orchards were an appealing choice. These areas were also home to a
large number of pigs
which increased the chance of transmission from bat to pig and as a
knock on effect, from
pig to human in the initial outbreak (Bengis et al. 2004). The
subsequent 2001 Indian and
Bangladeshi outbreaks of Nipah virus were assumed to be caused by
the ingestion of fruit
and fruit related products that had been contaminated with fruit
bat saliva and urine (WHO
2004).
Hantavirus Pulmonary Syndrome (HPS) is an infectious respiratory
disease which is endemic
to the Americas. The natural host of HPS is thought to be rodents,
and is typically carried by
the deer mouse (peromyscus maniculatus) in the United States.
Transmission from wildlife
13
to humans is through the respiratory route as a result of airborne
dispersion of rodent
excretions (Bengis et al. 2004), and a study carried out in 1998 of
177 cases of HPS in 29
states, estimated the casefatality proportion to be 45% (Young et
al. 1998). Outbreaks of
Hantavirus in humans in the United States have frequently been
linked to changes in
environmental conditions which are highly favourable to the rodent
populations (Epstein
1995). The HPS epidemic in the American Southwest of 1993 followed
an abundance of
rainfall increasing the food sources available for the deer mice.
The high population
densities driven out of their burrows by flooding increased the
chance for the virus to
flourish and transmit within rodent populations and eventually onto
human populations
(Epstein 1995) which can be likened to the perturbation effect
(Tuyttens et al. 2000; Carter
et al. 2007). Human activities such as herding livestock and
cleaning out rarely used rodent
infested areas have been attributed to increasing the risk of human
exposure to HPS
through contact with infected urine and faeces (Armstrong et al.
1995).
Avian Influenza (perhaps the highest profile wildlife disease
threat given volume of past
epidemics and the risk of a pandemic outbreak in the future (Swayne
2009)) is an infectious
viral disease found in birds (particularly wild water fowl) caused
by strains of the flu virus
not unlike the strains found in humans. In wild bird populations it
often causes no clinical
signs. Some forms of avian influenza have mutated in such a way
that they are able to
transfer from birds to humans populations. An example of this was
demonstrated in 1997,
when an outbreak of H5N1 (previously thought to only infect birds)
was first reported in
Hong Kong causing 18 cases of infection with a 33% mortality rate.
The clinical progression
in humans of this Influenza strain can be categorised into three
stages. The first stage of the
illness would usually show mild upper respiratory tract infection
and fever, or be
asymptomatic. The second stage is marked by additional symptoms of
severe pneumonia,
haematological liver and renal abnormalities. Finally the third
stage shows a highly
developed illness of acute respiratory distress syndrome, multiple
organ malfunction and
ultimately death (Tam 2002). The incidence of human infections
reported in all major
outbreaks of Avian Influenza to date has happened in people who
have high level of
interaction with poultry. It is thought these infections represent
direct birdtohuman
transfer of the virus, as fortunately there appears to have been
negligible humantohuman
spread. Nevertheless, the adaptation and mutation of these poultry
viruses could lead to a
new subtype of Influenza which is capable of sustaining itself
within a human population
alone. This is a key concern to public health services around the
world as the threat of a
14
highly contagious new pandemic could be potentially looming ever
closer (Bengis et al.
2004).
1.2 Wildlife disease and conservation Biodiversity can also be
affected by the emergence and spread of wildlife diseases. If
the
pathogen in question induces a high enough mortality rate in the
host, there is a risk of
losing entire populations of species and endangering many others;
disease induced
reductions in population size may significantly increase the chance
of local extinction due to
demographic fluctuations. Recent outbreaks have demonstrated this,
for example in
Australia the chytrid fungus Batrachochytrium dendrobatidis (Bd) is
thought to be behind
the fall in population size and perceivable extinction of at least
14 highelevation species of
rainforest frog (Retallick et al. 2004). This fungus is not unique
to Australia, and it’s affects
on the amphibian community can be felt worldwide. Using information
from a “last year
observed” database, Marca et al deduced that the frog genus
Atelopus has undergone 67
species extinctions since 1980 (Marca et al. 2005), which have been
generally attributed to
Bd. Bd has been categorised as an amphibiotic emerging infectious
agent (Daszak et al.
1999) and is considered to be pandemic. In 2008 the World
Organisation for Animal Health
reported Bd as a notifiable pathogen (OIE 2008).
The impact of pathogens on wildlife conservation can be damaging
when endangered or
threatened species are affected by an outbreak of an infectious
pathogen. The Tasmanian
devil population has continually declined by up to 90% from 1996
when a debilitating and
aggressive facial cancer tumour was first reported known as
Tasmanian Devil Facial Tumour
Disease (DFTD) (McCallum et al. 2007). This has caused the
Tasmanian Devil, which was
originally listed as “lowrisk” in terms of endangerment in 1996, to
be officially categorised
an “endangered” species as of 2008 by the International Union for
Conservation of Nature
(IUCN) Redlist (Hawkins et al. 2008). In the conservation biology
literature emphasis is
placed on the possible adverse consequences of wildlife disease due
to a decline in genetic
diversity (Epps et al. 2005; Schmid et al. 2009). The lasting
effect of a decline in genetic
diversity within a population is the reduction in the ability to
adapt to changes, for example
loss of habitat or fragmentation. A more immediate disadvantage is
the incapacity to resist
pathogen infection. DFTD is a prime example of how loss of genetic
diversity within
populations amplifies the risks posed by disease (Jones et al.
2004). This case highlights the
15
importance of surveillance and early action against emerging
diseases threats; as DFTD was
not quickly identified it has already spread across a large range
of the species which makes
eradication difficult (McCallum 2008).
Canine distemper (CDV) is a viral disease affecting the respiratory
tract, gastrointestinal
system, skin, reproductive tract, eyes, and nervous system. It is
classified as producing
clinical signs such as nasal discharge, transient fever, diarrhoea,
and weight loss. CDV has
been recognised as a pathogen of domestic dogs, however numerous
CDV infections of wild
species have been documented (Leisewitz et al. 2001). In 1994, the
Serengeti National Park
lion population were subjected to a devastating outbreak of canine
distemper. This led to
the overall population reducing in size by approximately onethird
(i.e. 1000 animals)
(RoelkeParker et al. 1996), a significant blow to the conservation
and protection effort of
the Serengeti National Park. However, as a consequence of intense
monitoring of the lion
prides, detailed observations were collected on the incidence of
CDV and their movement.
From this data it was deduced that it was unlikely lion movement
patterns could account
for the spread of the disease, which eventually led to the
exploration of other potential
reservoirs of the pathogen (Haydon 2008). It has been suggested
that the most probable
reservoir for CDV in the Serengeti lion epidemic was the domestic
dogs of the local villages.
Between 1991 and 1993, the seroprevalence of CDV increased in their
population which
preceded the 1994 outbreak in lions (RoelkeParker et al. 1996).
This 1994 outbreak of
canine distemper illustrates quite clearly the risks posed by
wildlife disease but also to the
great advantages of surveillance. It would have been impossible or
at the least very difficult
to determine how the outbreak evolved without such rich data
collection.
1.3 Wildlife diseases and the livestock industry
When a disease is classified as notifiable, it is required by law
that it is reported to
government authorities if discovered. This can lead to restrictions
placed on the movement
of livestock from affected premises and subsequently impact on the
“disease status” of a
country possibly leading to a ban on trade until the country is
considered “disease free”.
Therefore, there is potential for tremendous economic impacts when
wildlife disease can
transmit to livestock. Bovine TB (bTB), caused by Mycobacterium
bovis, is a focal point for
wildlife control because of the adverse consequences of the disease
on livestock production
16
(Donnelly & Hone 2010) and the significant affect it has on
trading as a result of EU trading
standards and procedures (Caffrey 1994; DEFRA 2011). bTB is
categorised as a bacterium
which causes chronic incapacitating disease in cattle, humans and
various wild species,
including the badger (Meles meles) (Bengis et al. 2004). During the
comprehensive
Randomised Badger Culling Trial (RBCT), designed to establish if
culling badgers reduced TB
in cattle, badgers were shown to be a source of infection to cattle
(DEFRA 2013). The trial
results suggested that culling of badgers over a fixed area of
150km2 would lead to an
average of 16% reduction in bTB incidence in cattle in the local
area. However, there are
also negative impacts of culling which can lead to an increases in
disease e.g. the
perturbation effect as mentioned previously (Prentice et al. 2014).
This situation has put a
strain on the economy as infected cattle have decreased production
of milk and infected
carcasses will be seized at abattoirs if detected (Amanfu 2006;
Firdessa et al. 2012). Known
infected cattle herds are put on a trade lockdown as required by
the EU trading standards
and are prohibited from moving until bTB free status is achieved.
There are also added
economic problems which come with the slaughtering of infected
cattle. It is estimated that
approximately 28,000 cattle were slaughtered in 2012 through bTB
infection (DEFRA 2012)
and the burden of compensation falls on the government or else must
be absorbed by the
individual infection sites which could result in farms going out of
business if the financial
impacts of the cull are too great.
Schmallenberg is an emerging livestock virus that has been detected
in parts of Europe and
the UK and it is transmitted via vectors such as mosquitoes and
midges to the livestock
hosts (DEFRA 2012) and wild ruminants (ECDC 2012). The virus is
characterised by the host
showing clinical signs including a decrease in milk production,
watery diarrhoea, and
occasional fever (Elbers et al. 2012), and there have also been
reports of congenital
malformations in ruminants (van den Brom et al. 2012). DEFRA has
stated that the most
likely cause of Schmallenberg in the UK is due to infected midges
being blown across the
channel, most probably from France (Conraths et al. 2013). Although
this is not a notifiable
disease as yet, it is being closely monitored and any farmers who
encounter the disease or
suspicious symptoms are advised to contact their local vet. Impact
assessments by the EFSA
(2012) and Harris (2014) suggest that if this virus were to spread
further across the UK and
Europe it is likely to have serious adverse economic impacts on the
farming industry (EFSA
2012; Harris et al. 2014) as meat and milk production could be
badly hampered.
17
Foot and mouth disease (FMD) is an infectious virus affecting
ruminants and is
characterised by clinical signs including loss of appetite, sudden
death of young, lameness,
blisters and reduced milk yield. Although recent outbreaks have
been predominantly in
livestock, wild ruminants can also play a role in the introduction
and spread of FMD (Condy
et al. 1969; Ward et al. 2007). The disease can have severe
consequences for animal health
and the economics of the livestock sector as exemplified by the
2007 outbreak in the UK.
During August and September 2007, FMD caused large disruptions to
the farming sector
and cost hundreds of millions of pounds in control efforts and
slaughtered animals (Cottam
et al. 2008). For example in Scotland there was an export ban on
live animals imposed until
the close of the year, the effect of this was a reduction of market
prices, although the real
measure of this depends on the initial state of the market before
the outbreak. This
represented a considerable cost to farmers of livestock and in turn
also represented losses
to the overall agricultural supply chain (Scottish Government
2008). The burden of a
restriction on movement requires farmers and other branches of the
agricultural supply
chain to diverge from the usual procedure and these effects can be
exacerbated if farmers
keep their stock for a longer period because of lower market
prices, all of which
demonstrate extra cost and strain on the livestock economy.
1.4 Wildlife Disease Surveillance
Surveillance is the first line of defence against wildlife disease
and the threats it can pose.
Wildlife disease surveillance aims to limit and end outbreaks of
disease before they have
the ability to cause major damage to public, livestock and wildlife
health (Belant & Deese
2010) . When used in an efficient and comprehensive manner,
surveillance can be
instrumental in controlling and overcoming disease outbreak (FAO
2011). There is an
increasing recognition of the necessity of wildlife disease
surveillance (Jebara 2004; Kuiken
et al. 2011). However, there are a range of issues associated with
surveillance of wildlife
disease, e.g. poor knowledge of basic ecology and distribution of
host species, this makes it
particularly challenging even when compared to surveillance in
livestock and humans.
The wildlife disease surveillance strategies currently undertaken
in Europe are few and far
between (Artois et al. 2001). The protocols for these activities
are still informal and as of yet
there is no structure in place to facilitate coordinated
surveillance/reporting of wildlife
disease between countries (Kuiken et al. 2011). As previously
mentioned, there is an
18
increasing acknowledgement that greater priority should be placed
on wildlife disease
surveillance, driven in part by the numerous examples of zoonotic
outbreaks in the human
population (SARS, Swine Flu, Avian Influenza etc). Leading
international wildlife
organisations and influential veterinary editorials have
highlighted the importance of
wildlife surveillance effort. The director general of the world
health organisation for animal
health (OIE) asserted that “Surveillance of wildlife diseases must
be considered equally as
important as surveillance and control of diseases in domestic
animals” as well as concluding
that the surveillance of wild animal disease is essential (Vallat
2008). In the first EWDA
meeting for wildlife health surveillance on 15th October 2009 in
Brussels, 25 representatives
presented summaries of the wildlife health surveillance in their
respective countries. Based
on these summaries Kuiken et al. (2011) showed that there are
significant differences in
surveillance approach across Europe. The number of surveillance
schemes in action per
country, the intensity of those schemes, the number of animals
examined, effort in terms of
the number of people employed, and the sources of funding vary
greatly. The authors
categorised the differences in surveillance in each country by
three different levels; no
general surveillance, partial general surveillance and
comprehensive surveillance, of which
the UK fell into the later. As a result of this, there was a
consensus among the participants
that wildlife health surveillance in Europe would profit from a
more formal network of
people actively contributing to this research area (Kuiken et al.
2011).
When undertaking surveillance of a given population (i.e. a group
of individuals of the same
species interacting within some defined area), the task becomes a
lot easier if the
population is closely managed (i.e. livestock), but unfortunately
this is not the case when
considering wildlife populations. Compared with surveillance in
livestock there are many
added complications to wildlife disease surveillance, for example
locating the population,
estimating accurately the size of the population and understanding
the demography and
transmission dynamics of the population including interactions with
other populations and
the wider community (i.e. other species). These added complications
can make it difficult to
obtain the samples required of a successful surveillance system
(Nusser et al. 2008). Many
wildlife surveillance strategies depend on methodologies based on
protocols developed for
livestock systems. However, as discussed, wildlife populations are
considerably more
complex and to date there has not been a detailed exploration of
whether methods used in
livestock are suitable for application to wildlife
populations.
19
Wildlife disease surveillance can be characterised under two broad
categories, active
surveillance and passive surveillance. Passive surveillance can be
generally defined as the
discovery and testing of naturally occurring deceased hosts (i.e.
animals that have not died
for the initial purpose of surveillance). There are instances of
routine collection of hunter
killed samples and collection of roadkill animals, but a primary
difficulty of passive
surveillance is that the strategy generally relies on members of
the public identifying and
delivering a case for diagnostic testing (Rhyan & Spraker
2010). The potential of passive
surveillance is hard to realise in practice since disease detection
is quite frequently time
sensitive (i.e. sensitivity of diagnostic tests may reduce sharply
with time since death) and
the incentive for the general public to report a case is relatively
low. There is also a
considerable chance of bias in the sample when relying on passive
surveillance especially if
the hostpathogen dynamic features significant disease induced
mortality or if behaviour of
infected individuals reduces or increases the chance that deceased
individuals are
encountered. Active surveillance in the context of this thesis is
defined as the capture and
subsequent testing of individuals driven by surveillance related
objectives. A primary
difficulty with this type of surveillance is that it can be more
costly than other options and
only limited funding is available (Lancoua et al. 2005). In
livestock systems active and
passive surveillance strategies are much simpler to implement, and
as highlighted
previously wildlife systems are much more complex. There are
numerous complications
when gathering samples in the field, including dynamic aspects of
population turnover,
habitat effects on density and distribution in space and time,
behavioural aspects affecting
sampling e.g. elusive nocturnal species trap shyness of animals
etc. These complications
typically result from dynamic processes which are subject to
stochastic fluctuations making
it more difficult to design and implement randomised sampling
strategies.
In summary, both passive and active surveillance have the potential
to be effective tools for
wildlife disease research, but they can suffer shortcomings
including underreporting, and
difficulties in designing effective surveillance strategies due to
the complexity of host
pathogen systems (Stallknecht 2007). There is therefore a need to
address these problems
systematically; in particular there have been calls for improved
panEuropean mechanisms
including defined standard protocols and data sharing (Genovesi
& Shine 2002; Kuiken et al.
2011; “WILDTECH Report Summary” 2014). This is a long term goal
which would aim for a
coordinated approach to surveillance and monitoring, to offer
increased protection from
disease outbreaks and incursions. For this to work, unquantifiable
biases need to be
minimised (e.g. human behaviour in passive systems) and/or
accounted for in subsequent
20
analysis. For the purposes of this thesis, we investigate only
active surveillance as the
characterisation of biases in passive surveillance requires a focus
on specific surveillance
scenarios and here we wish to explore generic aspects of wildlife
disease surveillance. For
example, results not shown indicate the ability of passive
surveillance to detect disease
depends strongly on the level of disease induced mortality and the
rate at which the
animals decay.
The key statistics we subsequently use to characterise the
performance of wildlife disease
surveillance systems are reviewed here. There are several
statistics that can be estimated
using surveillance information which give valuable information
about the population itself,
disease status and surveillance efficacy. However, in this thesis
there are two primary
statistics of interest used to investigate surveillance performance
and the effect of
population demography and disease transmission on surveillance
efficacy, the estimate of
prevalence and the probability of detection.
1.4.1 The Probability of Detection
The simplest and most widely used approach to estimating the
probability of disease
detection is to assume constant prevalence p and an effective
infinite population size (i.e.
assume sampling is with replacement/the population size is not
finite). These assumptions
lead naturally to a binomial formula for the probability of
detecting disease from n samples.
= 1 − (1 − ) (1)
a = the probability of detecting disease
p = the prevalence
The above formula can be used to carry out a power calculation as
follows. Rearranging
equation (1) in terms of n, gives an estimate of the sample size
required to obtain a
= ( )
( ) (2)
21
This required sample size increases rapidly as prevalence tends to
0, as demonstrated in
Figure 1.1. This can lead to “over sampling” especially for smaller
populations, since the
underlying assumption is that the population is infinite. To
counteract this effect which
would lead to the repeated sampling of individuals, a number of
authors (Martin et al.
1987; Artois et al. 2009a; Fosgate 2009) have considered modifying
this approach based
upon the hypergeometric distribution which accounts for finite
population sizes. This
approach leads to the following sample size calculation
= 1 − (1 − )/ × [ − 0.5( − 1)] (3)
where N = the total population size
D = the total number of diseased individuals within the
population
p =
= prevalence
Figure 1.1 shows equation (3), demonstrating the effect of
modifying equation (2) to
account for finite population sizes. Without this modification, at
low prevalence, the sample
size required to detect disease presence can be greater than the
population itself. As noted
above this would entail sampling the same individuals more than
once which is clearly
inefficient and given perfect tests completely unnecessary, at
least if the disease status of
individuals is assumed not to change over time. This is of course
consistent with the
assumption of unvarying prevalence on which both equations (1)(3)
rely. The hyper
geometric correction implemented in equation (3) ensures the
maximum sample size
required is capped at the total population size.
22
Figure 1.1: Effect of disease prevalence on the sample size
required to detect disease.
Plots are shown for varying levels of diseased individuals when the
probability of detection
a = 0.5 for both equation (2) and (3). Plot 1.a shows the effect of
number of diseased
individuals on sample size requirement described by equations (2)
and (3) for a fixed
population size of 100 (i.e. maximum prevalence is 0.1). Plot 1.b
shows the effect of number
of diseased individuals on sample size requirement from equations
(2) and (3) for a fixed
population size of 1000 where again maximum prevalence is
0.1.
Equations (3) can be rearranged to give an equation for the
probability of detection for
finite sized populations analogous to equation (1):
= 1 − 1 −
(4)
Note that equation (1) is the form that will be used herein when
referencing the binomial
equation for the probability of detection.
1.4.2 Estimating the True Prevalence: Bias and Standard
Deviation
In addition to simply detecting the presence of disease,
surveillance may also be called
upon to accurately estimate the prevalence in a population.
However, as we show in this
Equation (2)
Equation (3)
23
thesis achieving this can be quite difficult in the face of
demographic fluctuations within the
population. In contrast the standard approach is to ignore such
fluctuations and assume
constant prevalence p and population size. This leads to the
conclusion that the number of
infected individuals in a sample of n individuals, from a
population with prevalence p, is
drawn from a binomial distribution with Bin(n,p). Which has mean np
and variance
np(1-p). Therefore under these assumptions we find that the
binomial estimate of
prevalence is np/n =p and therefore
E[surveillance estimate of prevalence]= E[true prevalence]
(5)
i.e. the bias of the surveillance estimate (under the above
assumptions) is equal to 0.
Given the variance in the binomial estimate of the number of
infected cases in a sample of
size n, the standard deviation (std dev) in the corresponding
estimate of prevalence from
surveillance is:
as before, p =prevalence and n = sample size.
As with the probability of detection in equation (4), there can be
corrections made to
equation (6) to account for the finite size of the population.
Frequently in survey research,
samples are taken without replacement and from a finite population
of size N. In this
instance, and especially when the sample size n is proportionally
not small (i.e. n/N > 0.05),
a finite population correction factor (fpc) is used as a prefactor
on the right hand side of
equation (6) to define both the standard error of the proportion.
The finite population
correction factor is expressed as:
= − − 1 (7)
Figure 2 show the bias and standard deviation using the binomial
theory based on the
assumptions of constant population size and prevalence. As stated
above, the binomial
24
equation does not predict any bias in the surveillance estimate of
prevalence as the
predicted prevalence is equal to the true prevalence, p. However,
the error in this estimate
is nonzero and varies with true prevalence, as can be seen in
Figure 2.b. Note that the
correction factor in equation (7) is less than 1 for any N>2 and
sample size n>1, and
therefore the error in the estimated prevalence shown in Figure 2.b
will be reduced when
accounting for the finite size of populations.
Figure 1.2: Using binomial theory to estimate the expected bias and
std dev in the
surveillance estimate of prevalence. Plots are shown for sample
size n = 10 and varying
levels of prevalence in the population for equation (5) and (6).
Plot 2.a shows the predicted
bias in the prevalence estimate from surveillance. Plot 2.b shows
the predicted std dev in
the prevalence estimate from surveillance.
1.4.3 Improving wildlife disease surveillance
In recent years, in line with the heightened interest in wildlife
disease discussed above, a
number of authors have identified a need for both improved
implementation and
methodological developments to enhance the design and evaluation of
wildlife disease
surveillance (Stallknecht 2007; Hadorn & Stärk 2008; Artois et
al. 2009a; RyserDegiorgis
2013). In order to effectively do this, it is important to consider
the ecology of the
population under surveillance as this will have an impact on the
results obtained (Béneult et
al. 2014). A strategy that worked for one type of natural
population may not necessarily
work as well in another. Understanding how the dynamics of the
hostpathogen interaction
affect the efficacy of surveillance is key, and potentially the
most important step towards
improvement of surveillance systems. However such effects have yet
to be systematically
considered in the literature. There have, however, been attempts to
improve wildlife
0.0 0.2 0.4 0.6 0.8 1.0
0 .0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .0
0 .1
0 .2
0 .3
0 .4
0 .5
disease surveillance design by incorporating weighting schemes
based on habitat suitability
of the observed population (Nusser et al. 2008; Walsh & Miller
2010). However, there are
also other factors that should be taken into consideration if
required, for example dispersal
of populations, population fluctuations, disease stability,
seasonality and environmental
change. In this thesis we use mathematical modelling to explore how
key ecological
processes that govern wildlife populations, in particular
demographic fluctuations,
stochastic disease dynamics and spatial heterogeneity, impact on
surveillance. Simulation
modelling could make a positive contribution to this area, as
testing out scenarios in the
field is either very monetarily costly and/or time consuming or
altogether unfeasible. By
running simulations, what is expected from different surveillance
strategies in a wide range
of hostpathogen systems can be explored. Using such studies, it
would be possible to
better understand the results obtained from a given surveillance
strategy.
1.5 Modelling Wildlife Populations, Disease and Surveillance
Mathematical modelling is a tool, or more correctly a set of tools,
used to represent
different mechanisms in the natural world, and in particular
enables prediction of system
level impacts that results from interactions between multiple
mechanisms (LucioArias &
Scharnhorst 2012). Such techniques have a long established role in
mathematical biology.
Translating descriptions of biological processes, such as
hostpathogen interaction, into
mathematical language is beneficial for many reasons, for example,
the precise
mathematical language aids in formulating ideas and recognising
underlying assumptions as
well as utilising mathematical techniques to manipulate equations
in order to gain insights
into the behaviour of the modelled system. Unfortunately
nonlinearities and population
heterogeneities, which are of key importance in many biological
systems, make such formal
analysis of associated mathematical models difficult and typically
intractable. However a
key advantage of this approach to biological research is the use of
computers in performing
calculations and running simulations. This is a considerable time
saver and also enables the
exploration of many scenarios that would be unfeasible to study in
the field. There are,
however, compromises to be made with this approach. As with most
natural interacting
systems, the dynamics of a hostpathogen interaction are potentially
extremely complicated
and it is important to identify the key elements to include within
the model since
encompassing every aspect is typically unfeasible and such a
comprehensive approach is
unhelpful in terms of generating insights/understanding into how
the studied system works.
26
“In principle models should be developed from the simple to the
complex” (Murray 2002)
and this ethos has been adopted throughout this thesis. With every
added complexity
introduced to the model description, the longer and more intensive
are the methods
required to handle the equations. Insights obtained from relatively
simple model structures
are more generic and can also aid in the understanding of more
complex models.
1.5.1 Temporal Modelling of Wildlife Populations with Disease
When simulating natural populations it is essential to account for
key demographic
processes in the model, e.g. births, deaths and immigration. An
example is shown in Figure
1.3 which illustrates a population subject only to birth governed
by a logistic growth rate. In
this thesis, a primary interest is that of pathogen transmission,
and we model both primary
and secondary transmission. Primary transmission occurs when a
susceptible from the
modelled population becomes infected by routes including, infected
water sources,
contaminated food, and transmission from an individual outside the
modelled population.
Secondary transmission occurs when a susceptible individual from
the modelled population
becomes infected through contact with an infected individual within
the modelled
population. Vertical and pseudo vertical transmission represent the
infection of offspring
and young individuals by parents. However, unless such transmission
is infallible vertical
and pseudo vertical mechanisms are not capable of sustaining
infection and in this thesis
we do not consider such routes. Nonetheless such mechanisms could
easily be incorporated
as part of the models presented here. As well as these elements
describing demographic
processes and disease dynamics it will be important to model key
aspects of the process
surveillance in order to assess how it performs in differing
circumstances.
Compartmental models have been used frequently to describe
hostpathogen systems in
wildlife disease research (Renshaw 1991; Murray 2002) as they aim
to reduce the
complexity of hostpathogen dynamics into a manageable number of
disease status
“compartments”. There are many different examples of such
compartmental models; some
of the more wellknown include SI, SIS, SIR, SIRS and SEIR. In these
compartmental models S
represents a susceptible state, I an infected state, E a latent
state (i.e. infected but not yet
infective), and R represents a recovered state (recovered from
disease and no longer
infectious or able to be infected). There are two broad categories
of mathematical
modelling which can utilise such compartmental structure,
deterministic and stochastic. In
both cases the model state space typically represents the number of
individuals in each
category. Deterministic models ignore the random fluctuations that
can be observed in real
27
systems, solving standard differential equations simultaneously to
update the model at
discrete time steps. The deterministic nature of these models means
that, starting from a
given input they will always return the same output. Continuous
time stochastic models
incorporate random variation by utilising probabilistic equations
to determine a series of
events which update the model at randomly generated time intervals
and potentially (for
multiple event types) in a manner that is also stochastic in
nature. Such stochastic processes
naturally lend themselves towards not only population level
representations (e.g. where
numbers of susceptible and infected individuals are tracked) but
also to individual based
models where the disease status over time of each member of the
population is
represented. There are pros and cons for both deterministic and
stochastic approaches and
the differences between them are now demonstrated with a simple
birthdeath process SI
example.
A Simple Compartmental Model Worked Example
Consider the components of the state space S(t) and I(t) which
represent the number of
susceptible and infected individuals respectively at time t. Birth
is represented by logistic
growth defined by Verhulst (1845), and since this is a feature
embedded throughout the
models used in this thesis a more detailed explanation is required.
Deterministic logistic
growth is defined by the following equation:
where N is the total population (S + I), r the intrinsic growth
rate and K the carrying capacity.
Figure 3 demonstrates a population experiencing deterministic
logistic growth over time
until reaching the carrying capacity, K. All individuals born via
logistic growth are
susceptible, which implies no vertical or pseudo vertical
transmission.
S I
28
Figure 1.3: An example of a logistic growth curve A Plot is shown
to demonstrate how the
population increases over time using the logistic growth equation.
Despite an initial
exponential increase growth is ultimately limited by the carrying
capacity, K.
If there is also a per capita death rate, d, and a density
dependent secondary transmission
with contact rate . The deterministic approach then describes an SI
model with births and
deaths as:
Given that = +
The biological mechanisms underlying this model are summarised in
Table 1 below, which
shows the rate at which each event occurs as well as the associated
change in the
population. In the next section we describe how the basic model
description shown in Table
1.1 can be implemented as a stochastic model.
29
Table 1.1: SI model with rate and effects for each event type. A
table is shown which
contains the rates of all events which can occur at each time step
with the associated effect
on the state space. This model can be implemented as a discrete
state space stochastic
process or with a continuous state space, as a deterministic model
using ordinary
differential equations or an analogous system of stochastic
differential equations (see text
for details).
Figure 1.4 demonstrates how S, I and N population sizes develop
over time using both the
deterministic and discrete state space stochastic approach
(described below).
Figure 1.4: Deterministic and Stochastic Time Trajectories. Shown
here is the total
population (green), the susceptible population (blue) and the
infected population (red) over
time for both a deterministic and discrete state space stochastic
Gillespie implementation
of an SI model with births and deaths. The parameter values used
are = 1, = 0.06,
= 5 , = 50, and starting with initial conditions of S = 39, I =
1.
Event Rate Effect
S = S + 1
Death of S S = S – 1 Death of I I = I – 1
Infection S = S – 1 I = I + 1
0 5 10 15 20 25
0 1 0
0 1 0
30
As Figure 1.4 shows, the outcome of a stochastic and deterministic
simulation, although
following roughly the same pattern, can be quite different.
Deterministic modelling has its
merits, but in the interests of understanding the impacts of
variability on the dynamics of
hostpathogen systems, a stochastic approach has been used in the
models developed in
this thesis.
The mechanisms summarised in Table 1.1 can be used to formulate an
integer valued
discrete state space continuous time Markov process. This then
provides a natural
stochastic description of the assumptions implicit in equations 8
and 9 (made explicit in
Table 1.1). Under this Markov process, the probability that a given
event occurs during a
short time interval (t, t+ Δt ) is given by its rate multiplied by
Δt. The stochastic model can,
and in some of the models of this thesis will, be simulated using
the Gillespie algorithm
defined by Gillespie (1979). To run Gillespie’s algorithm, starting
from time t when the
system is in state x(t), the time to the next event, , is randomly
chosen from an
exponential distribution with parameter, (()), which is the total
rate (i.e. the sum
of the rates of all possible events) evaluated at time t. The
event, then occurs at time +
and is chosen from the set of possible events with probabilities
given by the rate of each
possible event divided by (()). The derivation of the exponential
waiting time
distribution for a Markov process which is the basis of the
Gillespie algorithm is as follows.
Let –
()= probability that no event has occurred up to time t ,
w hen starting at (0)at time t = 0.
Then considering the change in () over a small time gives,
+ | () = | () 1 − ()
Where to first order in , 1 − () = probability that no event has
occurred in
a small time interval (,+ ) i.e. 1 the probability that one of the
set of possible events
did occur.
31
Rearranging this equation and taking the limit as → 0 we find
( |(0))
≡ lim
⇒ | (0) = ()
This last line follows on noting that, by definition, nothing has
happened at time t=0 and so
0 | (0) = 1. Note that this equation forms the basis of the
Gillespie algorithm since it
shows that the time to the next event has an exponential
distribution.
For the model described in Table 1
= 1 −
+ + +
Thus starting at time t and the time is advanced to + with the
inter event waiting time
drawn from the exponential distribution. i.e., ~ exp () where ()
=
{(),()} represents the state space at time t. After the time to the
next event is
calculated, the event which has occurred is chosen randomly by
generating a number from
the uniform distribution, ~ (0, ). The next event is:
32
+ + +
Note alternative orderings of event types (and associated rates)
are allowed but the order
used does not affect the statistical properties of the model. The
state space is updated
accordingly, the rates are recalculated and the above process is
repeated until some
maximum time, , or an alternative stopping criteria, is reached. An
alternative
stochastic approach which ignores the discrete nature of
populations (and hence is closer in
spirit to the deterministic model) is that of stochastic
differential equations (SDE’s). The
Gillespie implementation, e.g. of the SI model described above, is
a continuous time
discrete statespace Markov process in which the number of infected
individuals (I) and
total individuals (N = S+ I) are represented as integer
variables.
Table 1.2 shows the expectation and variance of the updates that
would be obtained during
a small time interval from the Gillespie algorithm implementation
of the model based on
the description of events shown in Table 1.1. (i.e. from the
discrete state space SI model
with births and deaths). Comparison with Table 1.2 enables both
drift e.g. fN,B(X(t)) and
diffusion e.g. gN,B(X(t)) functions to be identified
However the SDE approach makes use of continuous variables to
represent the state space.
Using the simple compartmental example above, we can represent the
change in the
system state variables during an infinitesimally small time
interval dt as the following set of
stochastic differential equations:
() = , ()+ , ()+ ,()+ ,()
+ , () () + , () ()
+ ,()() + ,()()
The reader will notice that here we have chosen to represent the
dynamics in terms of the
numbers of infectives and the total size of the population rather
than using the number of
susceptibles and another variable. Therefore the state space at
time t is now represented
by the vector () = { (),()}. The quantities BB(t), BDS(t), BDI(t),
B2ry(t) are independent
Brownian motions corresponding to each of the four event types. For
small but finite dt the
quantities dBB(t), dBDS(t), dBDI(t), dB2ry(t) can be interpreted as
independent draws, from a
zero mean Gaussian with variance dt, for each event type and each
time point 0,dt,2dt, ...
,T(0,T). Thus e.g. E[dBB(t)]=0, E[dBB(t)dBB(t)]=dt and
E[dBB(t)dBDS(t)]=0. The so called drift
term , () represents the expected change in population size N
associated with the
birth event conditional on the system being in state (), whereas
the diffusion term
, () represents the variance in this update. There are analogous
drift and diffusion
quantities corresponding to each state variable for each event
type. These are detailed in
Table 1.3.
Table 1.2: Expectations and variance-covariance rates. Expectations
and variance
covariances in changes (during the time interval t to t+δt) to the
state space {I(t),N(t)}
associated with each event type in the discrete statespace model
described above for the
SDE implementation. All such quantities are shown to first order in
δt.
E- type
Event E[δN|X(t)] E[δI|X(t)] Var[δN|X(t)] Var[I|X(t)] Cov[δN,δI
|X(t)]
B Birth
0 0
DS Death of S − 0 0 0 DI Death of I − − 2ry Secondary
Trans- mission
Table 1.3: Expectation and variance-covariance. Expectation and
variancecovariances in
changes (during the time interval t to t+dt) to the state space
{I(t),N(t)} associated with each
event type in the SDE model as described above. All such quantities
are shown to first order
in dt.
The SDE implementation is the diffusion limit of the Gillespie
implementation if it is
constructed in such a way that the first and second order moments
of the stochastic
updates in the differential equations correspond with those of the
Gillespie
implementation. This ensures that the results are consistent
between the two
implementations. It is easy to see that this consistency is
achieved if the values of the drift
and diffusion terms in the SDE model are chosen by comparing Tables
1.2 and 1.3. This then
suggests, for example that , () = (1 − / ) and , () = (1 − /
)
with assignments corresponding to other combinations of event types
and state space
variables made in an analogous manner (i.e. matching up the entries
describing
expectations and variances in updates shown in Tables 1.2 and 1.3).
For events which
change both state space variables the above formulation also
ensures that associated
updates also have the correct covariance, between changes in N and
I.
There are pros and cons to both the SDE and Gillespie
implementations, for example the
Gillespie algorithm is computationally more intensive whereas using
SDEs is faster and
therefore facilitates more accurate estimation of model statistics
(i.e. a greater number of
realisations can be run). However, the discrete nature of the
statespace under the Gillespie
algorithm represents a more natural description of the population
and the processes that
affect it. In particular, it provides a more accurate
representation of population dynamics
for small populations.
E - type
Event E[δN|X(t)] E[δI|X(t)] Var[δN|X(t)] Var[I|X(t)]
B , () , () , ()
1.5.2 Spatial Temporal Modelling of Wildlife Populations, Disease
and Surveillance
As well as describing the temporal ecological interactions in
wildlife systems, mathematical
models can also include spatial temporal dynamics. Research has
shown how extrinsic
spatial heterogeneity (i.e. habitat and land use), has an impact on
disease prevalence and
persistence (Hagenaars et al. 2004). This finding is important in
terms of disease
surveillance and this is taken into consideration in practice by
targeting known habitats of
wildlife species (Nusser et al. 2008). However, as far as we are
aware, there has been no
research or practical application which has addressed how intrinsic
spatial heterogeneity
(i.e. as generated by demographic fluctuations and disease
dynamics) will affect surveillance
and the efficacy of surveillance strategies.
The implementation of spatial stochastic models used in this thesis
(see Chapter 4) builds
on the basis of a nonspatial model, incorporating dynamics and time
increments as
described in section 1.5.1. within a spatial metapopulation. Here
each node or “patch” in
the defined space updates through time via birth, death and
immigration events etc. Every
patch is connected in the spatial area by a distance kernel
describing the spread of disease
between patches and it is this mechanism which controls the spread
of disease from patch
to patch. The distance kernel decays with distance and thus limits
the extent to which each
patch can transmit disease. The closer the patches are, the more
likely they are to pass
disease to one another. Because every process added to the model
increases complexity,
the simulations become ever more computationally expensive. There
are many uses for
both spatial and nonspatial modelling approaches (Tilman &
Kareiva 1997), small scale
spatial heterogeneity is less significant (i.e. a single
population) and it is these instances
when nonspatial methods may be more appropriate. However, when
dealing with large
scale metapopulations, as can be seen in this thesis, spatial
heterogeneity is an important
factor to include.
1.6.1 Aims
The overall aim for this thesis is to investigate how attributes of
wildlife populations affect
surveillance efficacy. We focus primarily on the statistical
calculations of prevalence
estimation and the probability of detecting disease. This is
implemented in a generic
exploration and then subsequently with more specific
examples.
1.6.2 Thesis structure
Chapter 2
Chapter 2 uses a nonspatial stochastic simulation model to
implement a systematic
exploration of the effects of pathogen transmission and host
population dynamics on the
efficacy of disease surveillance systems. Our results suggest that
for the vast majority of
disease systems this leads to over confidence in terms of both the
power to detect disease
and the bias and precision of prevalence estimates obtained.
Accounting for such ecological
effects will permit improvements to surveillance systems and better
protection against
emerging disease threats.
Chapter 3
Chapter 3 utilises the results in Chapter 2 and applies these
findings to two worked
examples of disease systems in the wild: badgers and tuberculosis;
and rabbits and
paratuberculosis. We show that similar effects to those
characterised in Chapter 2 can be
seen in these disease systems and we explore other sources of
complexity and bias which
have the potential to affect surveillance efficacy. This
demonstrates the potential of the
nonspatial stochastic model to be used to quantify effects in real
systems and also shows
its potential as a tool to explore the potential impacts of known
or putative sources of bias,
illustrating the power of our approach to inform
surveillance.
Chapter 4
Chapter 4 extends the nonspatial model used in Chapter 2 and 3 to
explore spatial aspects
of wildlife disease and wildlife disease surveillance and their
subsequent effects on
surveillance efficacy. This chapter focuses on disease incursion
events, representing
emerging or reemerging disease threats, and in particular the
amount and extent of spatial
37
spread of disease in the system at the point of first detection by
the surveillance system.
Different spatial surveillance designs are considered and compared
to give a better
understanding of the key mechanisms driving surveillance
performance in spatial settings.
Chapter 5
Chapter 5 is a general discussion which brings the results from the
preceding chapters into
the wider research context.
Surveillance
This Chapter was originally written in the style of a paper with
the intention of submitting to
Ecology Letters. Submitted September 2014, First Author: Laura
Walton.
39
2.1 Abstract
We present the first systematic exploration of the effects of
stochasticity in pathogen
transmission and host population dynamics on the efficacy of
wildlife disease surveillance
systems. The design of wildlife disease surveillance currently
ignores fluctuations in these
processes. Our results suggest that for many wildlife disease
systems this leads to bias in
surveillance estimates of prevalence and over confidence in
assessments of both the
precision of prevalence estimates obtained and the power to detect
disease. Neglecting
such effects thus leads to poorly designed surveillance and
ultimately to incorrect
assessments of the risks posed by disease in wildlife.
Understanding such ecological effects
will enable improvements to wildlife disease surveillance systems
and better protection
against endemic, emerging and reemerging disease threats. Our
results suggest a need for
a wider exploration of the impacts of ecology on wildlife disease
surveillance.
2.2 Introduction
Surveillance is the first line of defence against disease, whether
to monitor endemic cycles
of infection (RyserDegiorgis 2013) or detecting incursions of
emerging or reemerging
diseases (Daszak et al. 2000; Kruse et al. 2004; Lipkin 2013).
Identification and
quantification of disease presence and prevalence is the starting
point for developing
disease control strategies as well as monitoring their efficacy
(OIE 2013). Knowledge of
disease in wildlife is of considerable importance for managing
risks to humans (Daszak et al.
2000; Jones et al. 2008) and livestock (Frölich et al. 2002;
Gortázar et al. 2007), as well as
for the conservation of wildlife species themselves (Cunningham
1996; Daszak et al. 2000;
Evenson 2008).
Recent public health concerns e.g. Highly Pathogenic Avian
Influenza (Artois et al. 2009b),
Alveolar Echinococcosis (Eckert & Deplazes 2004) and West Nile
Virus (Valiakos et al. 2014),
have heightened interest in wildlife disease surveillance (Vallat
2008) and led to a growing
recognition that current approaches need to be improved (Mörner et
al. 2002). For
example, there is no agreed wildlife disease surveillance protocol
shared between the
countries in the European Union (Kuiken et al. 2011). Furthermore
several authors have
argued that improvements are needed to the structure, understanding
and evaluation of
wildlife disease surveillance (Bengis et al. 2004; Gortázar et al.
2007).
40
Much current practice for wildlife disease surveillance (Artois et
al. 2009a) is based on ideas
developed for surveillance in livestock including calculation of
sample sizes needed for
accurate prevalence estimation (Grimes & Schulz 1996; Fosgate
2005) and detection of
disease within a population (Dohoo et al. 2005). A common feature
of these methods is that
they ignore fluctuations in host populations and disease
prevalence. These assumptions
lead naturally to sample size calculations (for both disease
detection and prevalence
estimation) and other analyses, based on the binomial distribution
and associated
corrections for finite sized populations such as the hypergeometric
distribution (Artois et
al. 2009a; Awais et al. 2009). Fosgate (2009) reviews current
approaches to sample size
calculations in livestock systems and emphasises the importance of
basing analyses on
realistic assumptions about the system under surveillance.
However, although constant population size and prevalence may often
be reasonable
assumptions for the analysis of livestock systems, they are
considerably less tenable in
wildlife disease systems that are typically subject to much greater
fluctuations in host
population density and disease prevalence. For example,
practicalities and changes in
population density make it much harder to obtain a random sample of
hosts of the desired
sample size in wildlife disease surveillance programmes (Nusser et
al. 2008) compared with
livestock systems. Furthermore the importance of temporal (Renshaw
1991; Wilson &
Hassell 1997), spatial (Huffaker 1958; Lloyd & May 1996; Tilman
& Kareiva 1997) and other
forms of heterogeneity (Read & Keeling 2003; Vicente et al.
2007; Davidson et al. 2008) in
population ecology and in particular their role in the dynamics and
persistence of infectious
disease has long been recognised (Anderson 1991; Smith et al.
2005). However, such effects
have yet to be systematically accounted for in the design of
surveillance programmes for
wildlife disease systems, or in the analysis of the data obtained
from them. Although there
have been some attempts to account for spatial heterogeneities in
the design of wildlife
disease surveillance by incorporating weighting schemes based on
habitat suitability of the
observed population (Nusser et al. 2008; Walsh & Miller 2010),
we are not aware of any
attempts to account for temporal fluctuations in prevalence or host
population size. Here
we address this gap by assessing the impact of stochastic
fluctuations in host demography
and disease dynamics on the performance of surveillance in a
nonspatial context.
We demonstrate analytically that correlations in fluctuations of
prevalence and population
density bias prevalence estimates obtained from surveillance.
Simulations, using logistic
models of population growth and susceptibleinfected disease
dynamics, support this
41
finding and further show that variation in prevalence estimates can
be considerably higher
than would be apparent from standard calculations based on constant
population size and
prevalence. We also explore the impact of fluctuations in
population density and prevalence
on the ability of surveillance to detect the presence of disease.
An approximate argument
indicates that, in comparison with the detection rate obtained by
assuming constant
prevalence, the true probability of disease detection is reduced by
fluctuations, and this is
confirmed by subsequent simulation. The potential range of possible
detection rates is
assessed by simulating a spectrum of hostpathogen systems at two
sampling levels to
demonstrate the potential range of performance that could be
expected when surveillance
is deployed in the absence of knowledge of the underlying wildlife
disease system.
2.3 Methods
The model represents a host population subject to demographic
fluctuations (births, deaths
and immigration) and the transmission of a single pathogen. At each
point in time t, the
statespace represents the total population size N(t), with I(t) of
these infected and
S(t) = N(t) - I(t) susceptible. In addition the prevalence is then
given by p(t) = I(t)/N(t).
The birth rate of individuals is logistic, rN(1 – N/k), with
intrinsic growth rate r and carrying
capacity k reflecting the assumptions that population growth is
resource limited. Individuals
have a per capita death rate μ and immigration occurs at a constant
rate ν.
A proportion γ of immigrants are infected, but otherwise all
individuals enter the
population (through birth or immigration) as susceptible since we
assume vertical and
pseudovertical transmission are negligible. Susceptible individuals
become infected
through primary transmission (contact with infectious environmental
sources including
individuals outside the modelled population) and secondary
transmission (contact with
already infected individuals from within the population). Primary
transmission occurs at
rate β0 S(t) while secondary transmission occurs at rate
βS(t)I(t).
Disease surveillance is incorporated into the model in the form of
active capture, testing
and release at per capita rate α for both susceptible and infected
individuals. All
surveillance testing is undertaken assuming perfect tests, which
means that our measures
42
of the performance of surveillance reflect a best case scenario. A
summary of this
conceptual model is given in Table 2.1 which shows all demographic,
epidemiological and
surveillance events with their corresponding rate and effect on the
statespace.
2.3.2 Statistics generated from the model
Since we allow immigration of susceptible and infected individuals
neither the population
nor the disease will become extinct and we therefore assume that
long term averages are
equivalent to ensemble expectations (typically approximated by
averages over many
realisations of the process). Each simulation is run for a period
of time to allow the
population to reach equilibrium before long run averages are
calculated. For example, the
expected mean E[N] and variance Var[N] of the population size are
recorded along with
the expected mean E[p] and variance Var[p] of disease prevalence.
Similarly other statistics
such as the covariance between the prevalence and population size
are calculated as
required.
During a so called surveillance bout individuals are captured at
per capita rate α, and both
the total number and the number of infected individuals caught are
recorded. Note this
could be easily extended to account for imperfect disease
diagnostics by recording the
number testing positive but here we assume perfect tests. When the
surveillance bout
ends, either because a target number of individuals has been caught
or because an upper
time limit has been reached, the sample prevalence is recorded. In
addition, if at least one
infected individual was caught we note that disease was detected.
Therefore over repeated
surveillance bouts it is straightforward to estimate the
probability of detection PD (the
proportion of bouts where disease is detected) and the mean
E[psurv] and variance
Var[psurv] of the prevalence estimates obtained from active
surveillance.
2.3.3 Model Implementation
The model is implemented as a continuous time continuous state
space Markov process,
based on a set of coupled Stochastic Differential Equations, SDEs
(see e.g. Mao 2007) and
simulated using the EulerMaruyama algorithm (see e.g. Higham 2001).
For details see
section 1.1 in Appendix 1. The model is also implemented as a
continuoustime discrete
state space Markov process (also described in section 1 in Appendix
1), which is simulated
using Gillespie’s exact algorithm (Gillespie 1976). The SDE
implementation has been
constructed so that it is the diffusion limit of the Gillespie
implementation. To achieve this,
the first and second order moments of the stochastic updates in the
differential equations
43
are chosen to correspond with those of the Gillespie
implementation, ensuring that the
results are consistent between the two implementations. The
Gillespie algorithm is
computationally more intensive whereas using SDEs is faster and
therefore facilitates more
accurate estimation of model statistics (i.e. a greater number of
realisations can be run) and
more extensive exploration of parameter space. However, the
discrete nature of the state
space under the Gillespie algorithm is a more direct implementation
of the model described
in Table 2.1 and provides a more accurate representation of
population dynamics for small
populations.
T
a
b
l
e
Table 2.1: Model structure. Event, Rate and Effect on the State
Space of the model.
Conceptually the effect of each event affects an individual and
this is reflected in the
discrete nature of the corresponding changes in the state space.
However, given this
underlying conception of the model there are a number of different
implementations which
can be considered including via the Gillespie algorithm and
stochastic differential equations
(see text for details).
Event Rate Effect
Birth (1 − /)) → + 1 Death of Susceptible → − 1 Death of Infected →
− 1 Susceptible Immigration
(1 − ) → + 1
Infected Immigration
2.4.1 Estimating Prevalence
In order to develop an understanding of the properties of wildlife
disease surveillance using
the above model we now develop expressions describing prevalence
estimates obtained by
continuous surveillance i.e. continuously deployed effort resulting
in per capita capture rate
α.
Consider the interval [0,T] during which the population history
is
[0,T ] = {(N(t ),p(t )): t [0,T]} where N(t) and p(t) represent the
population size and
disease prevalence at time t [0,T] respectively (see above). Let nT
represent the total
number, and iT the number of infected individuals sampled during
this time interval.
Conditional on the history [0,T ] the expectations of these
quantities are
E| [0,]= ∫ ()
.
Assuming perfect testing (as we do throughout this paper) the
surveillance estimate of
disease prevalence is simply the ratio psurv(T ) = iT/nT. If the
stochastic process
representing the disease system is ergodic, and given the inclusion
of immigration (see
above) we can rule out extinction, the long time limit of this
estimate can be equated with
its ensemble average (expectation over all histories) i.e.,
lim→ () = E[]= lim→
This can be reexpressed in the more suggestive form
E[]= E[()]+ [ (),()]
E[ ()] (1)
Thus when the covariance Cov[N(t),p(t )] = E[N(t )p(t )] - E[N(t
)]E[p(t )] between the
population size and the prevalence is nonzero the surveillance
estimate of prevalence is a
biased estimate of the true prevalence, E[p(t)]. Since
Cov[N(t),p(t)] will be zero when
either N(t) or p(t) are constant, this result leads to our first
and most important conclusion,
45
efficacy of surveillance.
2.4.1.1 Effect of host demography and disease dynamics
We now focus on surveillance estimates of prevalence based on
finite sample sizes, and
compare these to the continuous sampling theory prevalence estimate
(see equation 1).
Using the SDE implementation of the full model, Figure 2.1
illustrates how population
fluctuations and disease dynamics in the hostpathogen system affect
the efficacy of
surveillance (in terms of the bias and variance of estimated
prevalence). These results are
generated by simulating the system for a range of values of the
death rate μ, with other
parameters fixed. As the death rate increases the expected
population size decreases and
demographic fluctuations increase. For a given rate of disease
transmission β, increasing the
death rate reduces expected prevalence, and therefore simulating
for different values of μ
generates the range of prevalence values shown. Details of the
parameterisations used are
given in Table S.1.1 (see section 1.2 in Appendix 1). The resulting
relationship between
demography and expected prevalence for particular disease
characteristics (here a fixed
transmission rate, β) is illustrated in Figures 2.1.a and 2.1.b
which show increasing
population size and lower demographic fluctuations as expected
prevalence increases.
Simulations not shown here show that our results generalise,
holding for transmission rates
relative to a recovery rate (governing an additional transition
from I to S) and death rates
relative to birth rate, r.
Figure 2.1.c shows the bias in the surveillance estimate of
prevalence E[psurv] - E[p(t )]
obtained from the same set of simulations. Results shown are based
on 106 surveillance
bouts with sample size m = 10, where for each bout sampling is
conducted at rate α until
10 individuals have been caught and tested. The bias predicted by
continuous sampling
theory (which does not account for sample size) is also shown, and
in this case is extremely
accurate i.e. it agrees with simulation results. Figure 2.1.c shows
the bias in surveillance
estimates of prevalence for four different transmission rates. It
is important to note that the
results shown are conditioned on the underlying prevalence E[p(t)],
and therefore for a
given prevalence the populations associated with higher
transmission rates are more
variable than those with lower β. We therefore conclude that such
variability increases the
bias of surveillance estimates of disease prevalence. Finally,
Figure 2.1.d shows the
standard deviation in surveillance estimates of prevalence obtained
from the same set of
simulations. Comparison with the variability in prevalence
estimates expected under the
46
zero fluctuation assumption reveals that fluctuations in our
simulated wildlife disease
system considerably reduce the precision (increase the variance) of
estimates obtained by
surveillance. The variability of these estimates also increases
with demographic
fluctuations. Thus, the dynamics of the hostpathogen interaction
are integral in
determining the efficacy of surveillance in terms of prevalence
estimation.
Figure 2.1: Effect of host demography and disease transmission.
Data are shown for a
range of values of the death rate μ which controls the stability
and size of the population,
and thus determines disease prevalence for a given transmission
rate, β. For β=1 plot 2.1.a
shows that expected population size increases with expected
prevalence E[p(t)] (i.e. as μ
decreases) whilst plot 2.1.b shows that the coefficient of
variation of the population size
decreases. For the four values of β indicated and fixed sample size
m=10, plot 2.1.c shows
the bias E[psurv] - E[p(t )], and plot 2.1.d the standard deviation
in surveillance estimates of
prevalence, versus the expected value of true disease prevalence in
the system, E[p(t)].
Results shown are based on 106 surveillance bouts using the SDE
implementation of